## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 2051

As before , we

analytical expression for ( exp tÃ ) ( s ) that the calculation of the operator 7 - 1

exp tÂ involves the calculation of integrals of the form setsu e - a2tlul u1 . . . u du ...

As before , we

**assume**that aạt > 0 . It follows from equation ( 55 ) which gives theanalytical expression for ( exp tÃ ) ( s ) that the calculation of the operator 7 - 1

exp tÂ involves the calculation of integrals of the form setsu e - a2tlul u1 . . . u du ...

Page 2150

Since 0 ( x ) Şo ( T ) we have M ( S ) = M ( 80 ( T ) ) , and it may therefore be

disconnected , the closed set 8 is an intersection na de of spectral sets da . Now

clearly M ( 8 ) ...

Since 0 ( x ) Şo ( T ) we have M ( S ) = M ( 80 ( T ) ) , and it may therefore be

**assumed**, without loss of generality , that 8 go ( T ) . Since o ( T ) is totallydisconnected , the closed set 8 is an intersection na de of spectral sets da . Now

clearly M ( 8 ) ...

Page 2212

To prove ( ix ) it may be

follows from ( vii ) that 0 , = 0 , 02 = ' w and ... that is , we shall

functions fz and fy are not identically equal and one of them , say fa , differs from

both fy ...

To prove ( ix ) it may be

**assumed**that on = 0 , , for if z = E , & and w = Ezy itfollows from ( vii ) that 0 , = 0 , 02 = ' w and ... that is , we shall

**assume**that thefunctions fz and fy are not identically equal and one of them , say fa , differs from

both fy ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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### Common terms and phrases

adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero